Does the photon have a 4-velocity in a medium?

[PWiz]

You can take a collimated light beam, pass it through a lenses to expand the beam and then condense it back down again.

But is light really only moving through empty space in this setup? Won't the material the lenses are made of "delay" the photon in a manner previously discussed in this thread when the photon passes through them?

 

[Ibix]

But is light really only moving through empty space in this setup? Won't the material the lenses are made of "delay" the photon in a manner previously discussed in this thread when the photon passes through them?

With lenses, yes. But note that different parts of the wave-front experience different thicknesses of glass and different path lengths in free space (summing to the same optical path length). With diffractive optics (i.e. diffraction gratings), I don't think the answer is quite so obvious, since these can be plane structures. Although there's still some interaction with matter.

As I recall, the above linked paper by Giovanni et al describes a simpler setup than the one I described (fiber, free space, difraction grating, free space, fiber, from memory) and concludes that their experiment shows that the group velocity in free space is less than c. They provide the geometric argument ("some bits of the photon travel on a diagonal path") as a visualisation of why that's the case, and maths to show that the description matches their results. I suspect that whether the photon is "actually spreading out" or "actually has a lower group velocity when prepared this way" is a matter of interpretation - although I'm happy to be corrected on that.

 

[PFfan01]

As I recall, the above linked paper by Giovanni et al describes a simpler setup than the one I described (fiber, free space, difraction grating, free space, fiber, from memory) and concludes that their experiment shows that the group velocity in free space is less than c. They provide the geometric argument ("some bits of the photon travel on a diagonal path") as a visualisation of why that's the case, and maths to show that the description matches their results. I suspect that whether the photon is "actually spreading out" or "actually has a lower group velocity when prepared this way" is a matter of interpretation - although I'm happy to be corrected on that.

Constancy of photon speed in free space. According to the principle of relativity, Einstein light-quantum hypothesis, momentum-energy conservation law, and Maxwell equations are equally valid in all inertial frames. Thus as the carriers of light energy and momentum, any photons in free space keep moving uniformly after they leave a source observed in any inertial frames. On the other hand, observed far away from the source (especially at the infinity, which was used as an assumption to derive Doppler effect in the 1905 paper by Einstein), the light wave behaves as a (local) plane wave, while the photons for a plane wave move at the light speed in all inertial frames due to the invariance of Maxwell equations. From this it follows that the photons in free space move at the light speed in all directions independently of the motion of the source or the observer, which is the direct result from the principle of relativity.

 

[Dale]

Probably there is no phase four velocity, otherwise it would contradict the wave four vector, according to the recently-published paper. http://www.nrcresearchpress.com/doi/10.1139/cjp-2015-0167#.Vki8xGzovIU

This paper is conceptually a complete mess. It is a classical paper on classical light, but repeatedly talks about photons and gives them classical attributes. A lot of the justification of their concepts is based on this mostly-classical (and therefore mostly-wrong) picture of a photon.

 

[PFfan01]

This paper is conceptually a complete mess. It is a classical paper on classical light, but repeatedly talks about photons and gives them classical attributes. A lot of the justification of their concepts is based on this mostly-classical (and therefore mostly-wrong) picture of a photon.

From my understanding, the photon concept was introduced by Einstein’s light-quantum hypothesis, thus the photon energy cannot be solved by quantum theory because the whole quantum theory is developed based on the fundamental assumption: the Planck constant is a Lorentz invariant constant (Dirac) and the photon energy is equal to the Planck constant multiplied by frequency while the frequency is a pure classical concept. Just like the photon energy, the photon momentum in free space cannot be solved by quantum theory, because it is the direct result of the principle of relativity and Einstein’s light-quantum hypothesis. Thus I don’t think that paper “is conceptually a complete mess”.

PS: As we know, there is a serious contradiction between nonlocal indeterminacy of quantum theory and local reality of special relativity, specifically reflected in the superluminal propagation of quantum states of an entangled electron pair. In fact, there is another serious self-contradiction in quantum assumptions: As we know, the canonical momentum for an electron in a uniform magnetic field is not unique (indeterminacy), but its canonical momentum operator should correspond to an observable quantity according to the quantum assumptions. Let us forget those controversies.

 

[Dale]

Thus I don’t think that paper “is conceptually a complete mess”.

I do. I am still wading through it, but that is my opinion this far.

A single author is generally a bad sign for the quality of a paper. Occasional good papers are solo-authored, but more commonly such papers are of low quality. Usually it means that the author has never had their ideas seriously challenged and the writing has had insufficient internal review before submission.

PS, the rest of your post shows a lack of understanding of modern QFT, but seems off topic for this thread.

 

[PeterDonis]

I don’t understand what the “spatially structured” means.

It means that a "photon" is not what you think it is. See below.

a photon has no internal structure (it's not even a particle in the ordinary sense). I don't they're talking about that here though.

They are talking about the fact that in their experiment, the waves associated with the "photons" are not plane waves: as the abstract says:

"light beams have finite transverse size, which leads to a modification of their wave vectors resulting in a change to their phase and group velocities."

In other words, the word "photon" as they are using it does not mean "a particle of light"; it means "a wave packet of light whose properties are such that its group velocity is lower than c".

(Note that I can only read the abstract, as the full paper is behind a paywall, so I don't know if they are using classical wave optics or quantum wave optics to do their analysis. Either way, however, what I said above applies; in the quantum case, the "group velocity" would just be an expectation value instead of a classically calculated number.)

 

[PeterDonis]

From my understanding, the photon concept was introduced by Einstein’s light-quantum hypothesis

The original light-quantum hypothesis was Planck's, not Einstein's; he introduced it in order to derive a formula for black-body radiation that matched experimental data, which the classical Rayleigh-Jeans formula did not (the failure of the latter to do so was called the "ultraviolet catastrophe").

the whole quantum theory is developed based on the fundamental assumption: the Planck constant is a Lorentz invariant constant (Dirac) and the photon energy is equal to the Planck constant multiplied by frequency while the frequency is a pure classical concept.

No, you have it backwards. In quantum electrodynamics, the "frequency" of the photon is the energy. More precisely, the energy is the fundamental concept; the "frequency" is just an interpretation that arises when we take the classical limit. The same goes for momentum vs. wave number.

Let us forget those controversies.

These opinions about quantum theory are off topic for this thread.

 

[PWiz]

EDIT: Nevermind.

 

[Nugatory]

This thread started off on slightly shaky ground:

we know that the Fizeau experiment supports relativistic 4-velocity addition rule. But a recently-published paper says that the photon does not have a 4-velocity...I wonder who's right?

The answer to this question is that they're both right, and the apparent contradiction appears because a photon isn't what you think it is.

We often use the word "photon" in relativity discussions when we really mean "a pulse of light that we've localized to within the precision of our thought experiment so that we can speak as if it is at a single point instead of spread out through a region of space like any real pulse of light" - we do this because nobody wants to say, write, or read forty-three words when one word is available and will get the message across.

However, this convenient oversimplification leads to confusion and apparent contradiction when we come across something that is true of a pulse of light but not true of a photon, and that's what's happening here. A photon does not have a four-velocity, but there is a way of associating a worldline and a four-velocity to the light in Fizeau's experiment. If a photon were that pulse of light we'd have a contradiction, but it isn't. Now look at the informal blurb about that paper in Science:

A photon always moves at c?
From Einstein’s special relativity on down, the invariance of the speed of light in free space has been a central tenet of physics. Now, in a clever set of experiments, scientists in the United Kingdom have demonstrated that, in certain conditions, individual photons in free space can be slowed down to speeds measurably below the supposedly invariant light speed .

You'll see the same confusion there, shifting smoothly from the behavior of photons to the "supposedly invariant light speed" (and note that the abstract of the paper is more precise than the informal blurb and does not hint that the invariance of $c$ is only "supposed").

From my understanding, the photon concept was introduced by Einstein’s light-quantum hypothesis, thus the photon energy cannot be solved by quantum theory

The concept introduced by Einstein's light quantum hypothesis bears very little resemblance to the modern understanding of what photon is (for example, Einstein would not have hesitated to assign positions and velocities to his hypothetical light quanta). It is not altogether lacking in irony that Einstein's Nobel was awarded for the piece of his anno mirabilis work that has stood up least well to the test of the time.

It's easier to say what a photon is not than what it is, but that's a better discussion for the QM forum.

 

[Nugatory]

This seems like an abuse of terminology, does it not?

There is abuse going on, but it's the phrase "particle of light" that is abusive (or at least routinely misinterpreted).

The basic terminology problem here comes from the way that the word "particle" is used in quantum field theories for historical reasons (and because no one wants to be saying "quantized excitation of the <whatever> field" all the time). That meaning is so different from the standard English-language meaning of the word "particle" that confusion is almost inevitable when someone hears the term "particle of light" outside of a QFT textbook.

 

[PeterDonis]

This seems like an abuse of terminology, does it not?

How so? A wave packet is a perfectly well-defined notion, as is its group velocity, and it is what the term "photon" as it is used in the paper, as far as I can tell, is being used to refer to.

 

[PFfan01]

... In other words, the word "photon" as they are using it does not mean "a particle of light"; it means "a wave packet of light whose properties are such that its group velocity is lower than c".

(Note that I can only read the abstract, as the full paper is behind a paywall, so I don't know if they are using classical wave optics or quantum wave optics to do their analysis. Either way, however, what I said above applies; in the quantum case, the "group velocity" would just be an expectation value instead of a classically calculated number.)

Here there is a paper that presents Padgett-team experimental work in popular words:
“Photon Footrace: Slowing Down Light in Free Space”, by Stewart Wills, http://www.osa-opn.org/home/newsroom/2015/january/photon_footrace_slowing_down_light_in_free_space/#.Vk4HF2zovIW It says: “Now, in a clever set of experiments, scientists in the United Kingdom have demonstrated that, in certain conditions, individual photons in free space can be slowed down to speeds measurably below the supposedly invariant light speed (Science, doi: 10.1126/science.aaa3035).”

From the following link, you have free-access to Padgett-team original experimental report:
https://www.researchgate.net/profile/Daniel_Giovannini

 

[Ibix]

(Note that I can only read the abstract, as the full paper is behind a paywall

Arxiv: http://arxiv.org/abs/1411.3987

 

[PeterDonis]

n certain conditions, individual photons in free space can be slowed down to speeds measurably below the supposedly invariant light speed

And what do they mean by "individual photons"? It certainly isn't "a particle of light".

Furthermore, to the extent they use a 4-vector (the "wave vector") to describe the light, the vector is always null; that is obvious from their equation

$$
k_z^2 + k_x^2 + k_y^2 = k_0^2
$$

So there is no way of associating their "photon" with a 4-velocity, since that would be converting a null vector into a unit vector, which is impossible, as has already been pointed out in this thread. Of course, since the "photon" is moving in free space, not a medium, we would expect its wave vector to be null no matter what we think might happen in a medium. In short, this paper, interesting as it is, appears to be irrelevant to the topic of this thread.

(But if the wave vector is always null, what is all this about the "speed" of the photon being slower than c? That's because they are using "speed" to mean $k_z / k_0$, i.e., the longitudinal component of the wave vector divided by the timelike component. Or, to put it another way, they are using "speed" to mean the momentum of the wave along its direction of propagation divided by its energy. Because the wave is finite in spatial extent, i.e., it has non-zero transverse components to its wave vector, its "speed" defined in this way will be less than c even though the wave vector as a whole is null. But this "speed" is not the speed of a "particle" in any case.)

 

[PFfan01]

PeterDonis: The original light-quantum hypothesis was Planck's, not Einstein's; he introduced it in order to derive a formula for black-body radiation that matched experimental data, which the classical Rayleigh-Jeans formula did not (the failure of the latter to do so was called the "ultraviolet catastrophe").

Einstein light-quantum hypothesis rejected by Planck https://en.wikipedia.org/wiki/Albert_Einstein

In a 1905 paper, Einstein postulated that light itself consists of localized particles (quanta). Einstein's light quanta were nearly universally rejected by all physicists, including Max Planck and Niels Bohr. This idea only became universally accepted in 1919, with Robert Millikan's detailed experiments on the photoelectric effect, and with the measurement of Compton scattering.

 

[PeterDonis]

Einstein light-quantum hypothesis rejected by Planck

Yes, Planck rejected the hypothesis once he realized the full implications. That doesn't change the fact that Planck originally introduced the hypothesis, five years before Einstein, in order to derive the correct formula for black-body radiation. Planck was careful to use weasel words to the effect that the hypothesis wasn't claimed to be "real", just a convenient mathematical trick to get the right answer. But scientifically speaking, that's meaningless; scientifically speaking, he was the first to use the hypothesis, regardless of what he thought about it philosophically speaking.

 

[PFfan01]

Yes, Planck rejected the hypothesis once he realized the full implications. That doesn't change the fact that Planck originally introduced the hypothesis, five years before Einstein, in order to derive the correct formula for black-body radiation. Planck was careful to use weasel words to the effect that the hypothesis wasn't claimed to be "real", just a convenient mathematical trick to get the right answer. But scientifically speaking, that's meaningless; scientifically speaking, he was the first to use the hypothesis, regardless of what he thought about it philosophically speaking.

Planck’s energy-quanta hypothesis and Einstein’s light-quantum hypothesis

“In 1900, German physicist Max Planck calculated the observed distribution of radiation energy in blackbodies based on the assumption that the oscillating atoms in the walls of the blackbody do not emit radiation at all energies — only at highly prescribed values. This assumption leads to a very different, and correct, expression for the distribution of radiation energy in a blackbody. Planck’s assumption was based on a theory about the properties of atomic oscillations—not about the true nature of light. In solving another puzzle about electromagnetic radiation (see p. 15), Einstein later realized that light itself was quantized.”

Maurina Sherman, “Shedding Light on Quantum Physics”, Science & Technology Review, June 2005, pp.12-19
https://str.llnl.gov/str/June05/Aufderheide.html

 

[PeterDonis]

Planck’s assumption was based on a theory about the properties of atomic oscillations—not about the true nature of light.

I see the point, but I would want to check primary sources to confirm this; it's not what I recall from previous reading, but it's been quite some time since I looked at any sources from that period.

 

[Dale]

So I finished with that paper. His stance is definitely opposed to the concept of a photon having a four-velocity in matter. I don't know of a reliable reference that takes the opposite stance.

However, the paper does have its own weaknesses that reduce its credibility. So I wouldn't consider it definitive, but definitely suggestive that it does not.

 

[PFfan01]

So I finished with that paper. His stance is definitely opposed to the concept of a photon having a four-velocity in matter. I don't know of a reliable reference that takes the opposite stance.

However, the paper does have its own weaknesses that reduce its credibility. So I wouldn't consider it definitive, but definitely suggestive that it does not.

DaleSpam, good comments. A reliable source for the definition of four-velocity of light is the book by W. Pauli, Theory of relativity, (Pergamon Press, London, 1958), Eq. (14), p. 18. It is said that Einstein highly praised that book. So I would rather believe Pauli's book.

 

[Dale]

Does Paulis book assert that the four velocity of a photon is well defined in matter?

I am not sure why you are posting a weak paper when you have a strong textbook.

 

[PFfan01]

Does Paulis book assert that the four velocity of a photon is well defined in matter?

I am not sure why you are posting a weak paper when you have a strong textbook.

Because I like to read those papers which challenge mainstream views. I am a layman, but you are good expert. I would like to see how you rebut those non-mainstream views, and from this I can efficiently learn something. But seems you did not give any specific reasons why that paper "is conceptually a complete mess", and "does have its own weaknesses". I don't think "A single author is generally a bad sign for the quality of a paper." is a convincing argument. For example, the following famous retracted high-profile paper has 8 authors:

Retraction: Stimulus-triggered fate conversion of somatic cells into pluripotency

• Haruko Obokata,
  
• Teruhiko Wakayama,
  
• Yoshiki Sasai,
  
• Koji Kojima,
  
• Martin P. Vacanti,
  
• Hitoshi Niwa,
  
• Masayuki Yamato
  
• & Charles A. Vacanti

Nature 511, 112 (2014) doi:10.1038/nature13598
http://www.nature.com/nature/journal/v511/n7507/full/nature13598.html

Of course, it would be much more convincing if you have statistical numbers or cite references to support.

 

[Dale]

I don't think "A single author is generally a bad sign for the quality of a paper." is a convincing argument. ...
Of course, it would be much more convincing if you have statistical numbers or cite references to support.

You can even just look at this author's impact factor, eg on researchgate (http://www.researchgate.net/profile/Changbiao_Wang3/publications ). The average impact factor of his single authored papers is 1.5, while the average impact factor of his multiple author papers is 2.9.

Sure, you can find many examples of high quality single author papers, and many examples of low quality multiple author papers. But typically single authorship is associated with lower quality; in the case of this particular author almost a factor of 2 lower quality.

Regarding specific weaknesses of this paper:
Single authorship
Overly grandiose claims
Poor understanding of background literature
Mixing of quantum concepts into a classical paper
Questionable assumptions
Cumbersome notation

 

[PFfan01]

...

Regarding specific weaknesses of this paper:
...
Overly grandiose claims
...

Perhaps the most outrageous claim made by the author is a proof that Planck constant is a Lorentz invariant. No, not at all; Here it is the author himself who has made an implicit assumption.

 

[PFfan01]

Regarding specific weaknesses of this paper:
...
Poor understanding of background literature
...

The author claims that the Poynting vector does not necessarily represent EM power flow. I think this is incorrect. There are no experimental results indicating that the standard Poynting vector is inadequate. One might feel it strange that the photon momentum in moving matter has not the same direction as the flow of energy, but there is nothing wrong with it physically. This is simply a characteristic feature of the theory.

 

[PFfan01]

... Regarding specific weaknesses of this paper:
...
Questionable assumptions
...

This paper does not contribute anything worth publishing to the long going discussion on the Minkowski-Abraham problem. In particular, the critical analysis by the author of the PRL paper. It is obvious to all those that understand the meaning of special relativity that the Abraham photon momentum and energy in a medium cannot constitute a Lorentz four-vector. This fact, however, does not mean that the Abraham approach contradicts theory of relativity because the medium defines a preferred frame of reference. There is absolutely no reason why the motion of photons or other particles in a medium must look the same in all coordinate systems.

 

[Dale]

This fact, however, does not mean that the Abraham approach contradicts theory of relativity because the medium defines a preferred frame of reference. There is absolutely no reason why the motion of photons or other particles in a medium must look the same in all coordinate systems.

I agree here. What must be invariant is the outcome of any experiment.

In my mind this freedom to partition the total momentum into an EM part and a matter part is similar to the gauge freedom. The Lorenz gauge is indeed most convenient for relativity, but the Coulomb gauge is nonetheless a valid gauge which may be useful for certain cases.

 

[PFfan01]

... Regarding specific weaknesses of this paper:
...
Poor understanding of background literature
...

The paper claims that a “classical mathematic conjecture” is shown to be flawed in Ref. 41 (http://dx.doi.org/10.1139/cjp-2015-0198); however the so-called “classical mathematic conjecture” turns out to be a solid, well-established result of tensor calculus in textbooks.

 

[vanhees71]

I'm not sure what of the here discussed issues have to do with the question asked in the title of this thread.

The question asked in the title of the thread is very simple to answer. It is just unclear, what you mean by "4-velocity of a photon in a medium". This seems to indicate the usual misinterpretation of the word "photon" as a kind of point-particle concept. This is always misleading.

To treat a photon in the medium you need relativistic many-body QFT and to calculate the photon polarization tensor, which is not a trivial thing. The closest quantity you can get which is close to something like a four-velocity is the dispersion relation for photon quasiparticles in the medium. For a detailed treatment, see e.g.,

C. Gale, J. Kapusta, Thermal Field Theory, Cambridge University Press.

The other here discussed papers are about the age-old question of the definition of a covariant total four-momentum of electromagnetic fields, and this has a long history of confusion. I'm pretty confident that von Laue got it right. The reason is that for the interacting electromagnetic field, you cannot simply integrate the Belinfante energy-momentum stress tensor over the entire space of an arbitrary observer and expect to get a four-vector, because for this to hold true the corresponding four-current must be conserved, i.e., for an arbitrary inertial frame (let's discuss only the special relativistic case first) a tensor field $T^{\mu \nu \rho\ldots}$ leads to a four-tensor of lower rank via
$$\mathcal{T}^{\nu \rho\ldots}=\int_{\mathbb{R^3}} \mathrm{d}^3 \vec{x} T^{0\nu \rho \ldots},$$
only if
$$\partial_{\mu} T^{\mu \nu\rho\ldots}=0.$$
E.g., for electric charge you always must have $\partial_{\mu} j^{\mu}=0$, i.e., local charge conservation, due to gauge invariance and thus the electric charge is always
$$Q=\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} j^0$$
conserved and a scalar, i.e., independent over which space-like hypersurface you integrate provided you cover the entire charge of the system.

The Belinfante tensor is the correct gauge-invariant energy-momentum tensor of the electromagnetic field, because it also can be derived as the source of the gravitational field in Einstein's field equations from the electromagnetic field. However (even in special relativity), it is clear that in the case of an em. field interacting with charge-current distributions, it does not define an energy-momentum four-vector, because
$$\partial_{\mu} T^{\mu \nu} = -F^{\nu \rho} j_{\rho}.$$
Only the total energy-momentum tensor
$$\Theta^{\mu \nu}=T^{\mu \nu} + \Theta_{\text{matter}}^{\mu \nu}$$
is conserved, i.e., it fulfills
$$\partial_{\mu} \Theta^{\mu \nu}=0,$$
and the total energy-momentum
$$P^{\nu}=\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} \Theta^{0 \nu}$$
defines a proper (conserved) four-vector under Lorentz transformations.

The proof uses the four-dimensional Gauss integral theorem and was known for sure to Poincare and von Laue. There is a lot of confusion in the textbook literature about this, because often the authors forget this fundamental mathematics of tensor calculus and the integral theorems. Then you have all kinds of unnecessary nonsense debates about "hidden momentum" (there's no such thing but just mechanical and electromagnetic stress and the details of the famous formula $E=m c^2$, which implies that also stress adds to the total invariant mass of a composite system), the famous "4/3 problem" in the theory of charged extended bodies (charged classical point particles do not exist in the strict sense at all), etc. The only good thing about this is that you have nice examples to analyze within the correct machinery of tensor analysis, and von Laues books on relativity are masterpieces in doing right this.

Unfortunately, I cannot read the articles in the Can. J. Phys. because our University is not subscribed to this journal :-(.

 

[Dale]

The other here discussed papers are about the age-old question of the definition of a covariant total four-momentum of electromagnetic fields, and this has a long history of confusion. I'm pretty confident that von Laue got it right.

I find the position of Pfeiffer et al the most compelling.

http://arxiv.org/abs/0710.0461

They are not the first to recognize the correct resolution of this question, but I like their paper a lot.

 

[PFfan01]

... Regarding specific weaknesses of this paper:
...
Cumbersome notation

And the level of English is poor; I fear that most readers would be irritated by the quality of the English. Example of incomprehensible sentences:

“Maxwell equations support various forms of momentum conservation equations, which is a kind of indeterminacy. However it is this indeterminacy that results in the question of light momentum.”

 

[PFfan01]

... Regarding specific weaknesses of this paper:
...
Questionable assumptions
...

Questionable plane-wave model.

The article is to seek resolution of the question of light momentum in media. Frankly, I fail to see the problem - the "question" of what form to have for the momentum of a plane-wave in an (infinite) medium seems too academical to me. Since there are no plane waves, which are merely abstractions, a sometimes convenient linear basis to study wave propagation, how can I know if this "problem" is not coming from this mental construction which is an artifact?

 

[PFfan01]

... Regarding specific weaknesses of this paper:
...
Poor understanding of background literature
...

Poor understanding of the work of Pfeifer et al

In his introduction, the author presents a list of earlier works and, rather than present a scholarly and balanced assessment of these, seeks merely to discredit them with unsubstantiated comments.

The work of Pfeifer et al is belittled in the phase "Clearly, it is an insufficiency of the Pfeifer-coworkers theory that the EM momentum in a medium cannot be uniquely defined". Why "clearly"? The fact that the total momentum includes a contribution with both medium AND EM fields makes it quite clear that this is indeed the case, indeed the whole Abraham-Minkowski problem may be understood in precisely these terms.

 

[PFfan01]

... Regarding specific weaknesses of this paper:
...
Poor understanding of background literature
...

This paper relies on a major misunderstanding. Applying Abraham definition to a single photon propagating as a plane wave state through a dielectric medium, the author deduces that the result is not covariant and thus violates the relativity principle. But the momentum conservation law which follows from Maxwell equations, which are manifestly Lorentz covariant, certainly respects the relativity principle. The argument advanced by the author can at best show that Abraham momentum is frame dependent, but not that it is inconsistent with the relativity principle.